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Measures of noncompactness in the space $$L_ 0$$ and a generalization of the Arzelà-Ascoli theorem. (English) Zbl 0753.47022
The authors in this paper prove some inequalities between the Hausdorff measure of noncompactness and other numerical parameters. They verify the optimality of the coefficients, which appear in these inequalities. The main purpose of this paper is generalizing both of Fréchet-Šmulian compactness criterion and the Arzelà-Ascoli theorem.

##### MSC:
 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.